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Let
be a system of autonomous ordinary differential equation in
 defined by a vector field
 . A set  is said to be an attracting set[GH,P] if
 is closed and invariant,- there exists an open neighborhood
 of  such that all solution with initial solution in  will eventually enter  ( ) as
 .
Additionally, if  contains a dense orbit then  is said to be an attractor[GH,P].
Conversely, a set  is said to be a repelling set[GH] if  satisfy the condition 1. and 2. where
 is replaced by
 . Similarly, if  contains a dense orbit then  is said to be a repellor[GH].
- GH
- GUCKENHEIMER, JOHN & HOLMES, PHILIP, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer, New York, 1983.
- P
- PERKO, LAWRENCE, Differential Equations and Dynamical Systems, Springer, New York, 2001.
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"attractor" is owned by Daume.
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(view preamble)
| Also defines: |
attracting set, repelling set, repellor |
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Cross-references: orbit, dense, contains, eventually, solution, neighborhood, open, invariant, closed, vector field, ordinary differential equation, autonomous
There are 5 references to this entry.
This is version 1 of attractor, born on 2005-05-20.
Object id is 7089, canonical name is Attractor.
Accessed 3747 times total.
Classification:
| AMS MSC: | 34C99 (Ordinary differential equations :: Qualitative theory :: Miscellaneous) |
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Pending Errata and Addenda
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